Parametric curve - University of Washington

10.1/13.1 Parametric Curves Intro (2D and 3D)

Parametric equations: x = x(t), y = y(t), z = z(t)

To plot, you select various values of t, compute (x(t),y(t),z(t)), and plot the corresponding (x,y,z) points.

The resulting curve is called a parametric curve, or space curve (in 3D).

We also like to write the equation in vector form:

() = (), (), () = a position vector for the curve

i.e. if the tail of this vector is drawn from the origin, the head will be at (x(t),y(t),z(t)) on the curve.

Various Parametric Facts: 1. Eliminating the parameter (a) Solving for t in one equation, substitute

into the others. (b) Use (sin(u))2 + (cos(u))2 = 1.

This gives the surface/path over which the motion is occurring.

All points given by the parametric equations: x = t , y = cos(2t) , z = sin(2t) are on the cylinder: y2 + z2 = 1

z

x y

All points given by the parametric equations: x = tcos(t) , y = tsin(t) , z = t are on the cone: z2 = x2 + y2

z

y x

2. Intersection issues: (a) To find where two curves intersect, use

two different parameters!!! We say the curves collide if the intersection happens at the same parameter value.

(b) To find parametric equations for the intersection of two surfaces, combine the surfaces into one equation. Let one variable be t and solve for the others. (Or use sin(t), cos(t) if there is a circle involved)

3. Basic 2D Parametric Calculus

(Review of Math 124):

From

the

chain

rule

=

.

Rearranging gives

/ = / = slope.

Note that if y = f(x), this says

(())

=

(/()).

So the 2nd derivative satisfies:

()

=

(())

=

(

())

/

4. Vector Calculus

If () = (), (), (),

we define

() =

( + ) - () ( + ) - () ( + ) - ()

lim

0

,

,

so () = (), (), ().

We also define () = (), (), ().

In 13.3, we will see that () gives

information about the curvature.

In 13.4, we will see that () is a velocity vector, |()| is the speed, and ()is an acceleration vector.

We also define

() = (), (), () .

Morale, do derivatives and integral component-wise.

Ex: Consider () = , cos(2), sin(2). (a) Find (), |()|, and (). (b) Find (/4) and (/4). (c) Give the equation for the tangent line at = /4

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